These formulas come up again and again in assignments, B.Pharm and nursing papers, MPH exams and USMLE-style questions. For a diagnostic test, label the 2 × 2 table as true positives (TP), false positives (FP), false negatives (FN) and true negatives (TN).
| Measure | Formula | What it tells you |
|---|
| Sensitivity | TP ÷ (TP + FN) | Proportion of people with the disease who test positive |
| Specificity | TN ÷ (TN + FP) | Proportion of people without the disease who test negative |
| Positive predictive value | TP ÷ (TP + FP) | Chance a positive result is a true positive; depends on prevalence |
| Negative predictive value | TN ÷ (TN + FN) | Chance a negative result is a true negative |
| Positive likelihood ratio | Sensitivity ÷ (1 − specificity) | How much a positive result raises the odds of disease |
| Absolute risk reduction (ARR) | Control event rate − treatment event rate | The absolute difference a treatment makes |
| Relative risk reduction (RRR) | ARR ÷ control event rate | The proportional reduction in risk |
| Number needed to treat (NNT) | 1 ÷ ARR | How many people need treating to prevent one event |
| Standard error of a mean | SD ÷ √n | How precisely the sample mean estimates the population mean |
| z-score | (x − mean) ÷ SD | How many standard deviations a value is from the mean |
Worked example: a diagnostic test and a treatment effect (illustrative data). A new test is used on 100 people with a disease and 200 without. It gives 90 true positives, 10 false negatives, 20 false positives and 180 true negatives.
| Measure | Working | Result |
|---|
| Sensitivity | 90 ÷ (90 + 10) | 0.90 (90%) |
| Specificity | 180 ÷ (180 + 20) | 0.90 (90%) |
| PPV | 90 ÷ (90 + 20) | ≈ 0.82 (82%) |
| NPV | 180 ÷ (180 + 10) | ≈ 0.95 (95%) |
| LR+ | 0.90 ÷ (1 − 0.90) | 9.0 |
In a separate illustrative trial, 20% of the control group and 15% of the treated group have a heart attack. ARR = 0.20 − 0.15 = 0.05 (5 percentage points), RRR = 0.05 ÷ 0.20 = 25%, and NNT = 1 ÷ 0.05 = 20, so about 20 people need treatment to prevent one heart attack. Reporting both ARR and RRR avoids making a small absolute benefit sound large.